Dressel.Gruner.Electrodynamics of Solids.2003

We can solve this equation by assuming that the distribution function follows the spatial and time dependent perturbation of exp{i(q · r − ωt)} and has itself the form fk1(t) exp{i(q · r − ωt)}. Taking the Fourier components in r and t, we solve Eq. (5.2.10) for f 1, obtaining

The linearized distribution function yields a finite relaxation time τ and, consequently a finite mean free path = vFτ . The current density can in general be written as

in terms of the distribution function f 1(q, k, ω), where vk is the velocity of the electrons for a given wavevector k and the factor of 2 takes both spin directions into account. Using Eq. (5.2.11) for f 1(q, k, ω) we obtain

As J(q, ω) = σˆ (q, ω)E(q, ω), the wavevector and frequency dependent conductivity can now be calculated. Let us simplify the expression first; for cubic symmetry

where nE represents the unit vector along the direction of the electric field E. In the spirit of the Sommerfeld theory the derivative (−∂ f 0/∂E) is taken at E = EF, as only the states near the Fermi energy contribute to the conductivity. Second,

we consider this equation for metals in the limit T = 0, then the Fermi–Dirac distribution (5.2.8) is a step function at EF. With

the integral has contributions only at the Fermi surface SF. Under such circumstances we can reduce Eq. (5.2.14) to

¯

where we have also converted the volume integral dk into one over surfaces dS of constant energy as

5.2.2 The q = 0 limit

In the so-called local limit q → 0, the conductivity simplifies to

and we recover the familiar equation of the Drude model for the frequency dependent conductivity. The velocity term (nE ·vk)vk/vk averaged over the Fermi surface is simply 13 vF and therefore

statistics, only electrons near the Fermi energy EF are important, and therefore the mean free path = vFτ .

5.2.3 Small q limit

Next we explore the response to long wavelength excitations and zero temperature starting from Eq. (5.2.16). With the dc conductivity σdc = N e2τ /m, straightfor-

112 5 Metals

after some algebra we find that

there are no losses in the opposite, homogeneous, limit. The reason for this is clear: ω > qvF would imply a wave with velocity ω/q greater than the Fermi velocity vF; the electron gas clearly cannot respond to a perturbation traveling with this velocity.

5.2.4 The Chambers formula

The q dependent response is intimately related to the non-local conduction where the current density J at the position r is determined also by fields at other locations r = r . Here we develop an approximate expression for the current which depends on the spatial distribution of the applied electric field. Such a situation may occur in the case of clean metals at low temperatures when the mean free path is large. Let us consider an electron moving from a point r to another position taken to be the origin of the coordinate system. At the initial location, the electron is subjected to an electric field E(r) which is different from that at the origin. However, because of collisions with the lattice or impurities, the momentum acquired by the electron from the field at r decays exponentially as the origin is approached. The characteristic decay length defines the mean free path , and the currents at the origin are the result of the fields E(r) within the radius of = vFτ .

The argument which accounts for such a non-local response is as follows. When an electron moves from a position (r − dr) to r, it is influenced by an effective field E(r) exp{−r/ } for a time dr/vF. The momentum gained in the direction of motion is

in the following we drop the indication of the position (0). By integrating the above equation from the origin to infinity, the total change in momentum for an electron at the origin is found. Performing this calculation for all directions allows us to map out the momentum surface in a non-uniform field, and the deviations from a sphere centered at the origin constitute a current J. The following arguments lead to the expression of the current. Only electrons residing in regions of momentum space not normally occupied when the applied field is zero contribute to the current.

The density of electrons N moving in a solid angle d and occupying the net displaced volume in momentum space P is

where P is the total momentum space volume. The contribution to the current density from these electrons is

Substituting Eq. (5.2.25) into this equation and integrating over the currents given above yields

N e2 /(mvF). Equation (5.2.27) represents the non-local generalization of Ohm’s law for free electrons, and reduces to J = σdcE for the special case where → 0, as expected. The Chambers formula [Cha90, Pip54b] is valid for finite momentum, but as the Fermi momentum is not explicitly included its use is restricted to q < kF, and in general to the small q limit. A quantum mechanical derivation of Chambers’ result was given by Mattis and Dresselhaus [Mat58]. In Appendix E the non-local response is discussed in more detail.

5.2.5 Anomalous skin effect

In Section 2.2 we introduced the skin effect of a metal and found that the skin depth is given by

c

δ0(ω) = (2π ωσdc)1/2

in the low frequency, so-called Hagen–Rubens regime. In this regime we also obtained the following equation for the surface impedance

where the frequency dependence enters through the skin depth δ0 and through the free space wavelength λ0 of the electromagnetic field. The condition for the above results is that the mean free path is smaller than δ0 and hence local electrodynamics apply. At low temperatures in clean metals this condition is not met because becomes large, and the consequences of the non-local electrodynamics have to be

δeff

Fig. 5.9. Electron trajectories near the surface of a material (solid line) in the case of the anomalous regime, where the mean free path is much larger than the effective skin depth δeff (dashed line). According to the ineffectiveness concept, only electrons traveling nearly parallel to the surface are effective in absorbing and screening electromagnetic radiation when δeff, since the carriers moving perpendicular to the surface leave the skin layer before they scatter.

explored in detail. This can be done by using the Chambers formula (5.2.27), or alternatively the solution of the non-local electrodynamics, as in Appendix E.1.

An elegant argument, the so-called ineffectiveness concept due to Pippard [Pip47, Pip54a, Pip62] reproduces all the essential results. If the mean free path is larger than the skin depth, the effect on the electrons which move perpendicular to the surface is very different from those traveling parallel to the surface. Due to the long mean free path, electrons in the first case leave the skin depth layer without being scattered; the situation is illustrated in Fig. 5.9. Thus, only those electrons which are moving approximately parallel to the surface (i.e. their direction of motion falls within an angle ±γ δeff/ parallel to the surface of the metal) contribute to the absorption [Pip54b, Reu48]. The (reduced) number of effective electrons Neff = γ N δeff/ also changes the conductivity, which we write as:

where γ is a numerical factor of the order of unity, and can be evaluated by using Chamber’s expression [Abr72, Pip54b, Reu48]. The factor depends on the nature of the scattering of the electrons at the surface; for specular reflection γ = 8/9 and for diffuse reflection γ = 1. In turn the effective conductivity leads to a modified skin depth δeff = c/ (2π ωσeff)1/2, and by substituting this into Eq. (5.2.29) we obtain through selfconsistence for the effective skin depth

Fig. 5.10. Fermi sphere with radius kF corresponding to electronic states with momentum h¯ k and h¯ (k + q). The shaded region represents the momentum space to which single- particle–hole pairs can be excited with momentum h¯ q so that |k| < |kF| and |k+ q| > |kF| (only values of k where k is occupied and k + q is empty, and vice versa, contribute): (a) small q values, (b) intermediate q values, (c) large momentum |q| > |2kF|.

The minimum excitation energy is zero for |q| ≤ 2kF. However, for |q| > 2kF the minimum energy occurs for −q k and |k| = kF, and

finally leading to the following condition for electron–hole excitations to occur:

This region is indicated in Fig. 5.11 by the hatched area.

Next we want to evaluate the conductivity for intraband transitions (l = l ) starting from Eq. (4.3.32). In this case,

if we consider one single band. Here we have to split up the summation as

We have replaced k + q by k in the first term, which is allowed as the summation over k involves all the states, just as the summation over k+q does. We now replace the summation −1 #k by the integration over the k space 2/(2π )3 ! dk and, in the spirit of the relaxation time approximation, η is substituted by the relaxation

Wavevector q

Fig. 5.11. Excitation spectrum of a three-dimensional free-electron gas; the transferred energy is plotted as a function of transferred momentum. The pair excitations fall within the shaded area. In the region h¯ ω > (h¯ 2/2m)(q + 2kF)q, the absorption vanishes since h¯ ω is larger than energies possible for pair creation. For h¯ ω < (h¯ 2/2m)(q − 2kF)q we find σ1 = 0. Also shown is the dispersion of the plasma frequency ωp(q) calculated from

Eq. (5.4.28) using typical values for sodium: ωp = 5.9 eV and vF = 1.1 × 108 cm s−1.

rate 1/τ . Evaluating the matrix element for the conduction band in the case of a free-electron gas Ek = h¯22mk2 , we can rewrite the diagonal elements of Eq. (5.3.4) as [Lin54]

¯

The first square bracket is the matrix element | k + q|p|k |2 evaluated using Eq. (3.1.5) to consider the transverse components. For transverse coupling between the vector potential A and momentum p, we keep only those components of p